Finding the Discharge Intensity in a Wide Rectangular Channel with Alternate Depths

Given:

- Two alternate depths: 1.0 m and 1.5 m
- Width of the rectangular channel is not given

To find:

- Discharge intensity in the channel

Solution:

1. Understanding Alternate Depths

- Alternate depths are the two possible water depths in a channel which can carry the same flow rate or discharge.
- In other words, if there are two different depths in a channel and the same amount of water flows in both cases, these depths are alternate depths.

2. Calculation of Discharge Intensity

- To find the discharge intensity, we need the cross-sectional area of the channel and the velocity of the water.
- The cross-sectional area can be calculated using the alternate depths and the width of the channel.
- Let's assume the width of the channel is "b".
- The cross-sectional area (A) of the channel can be calculated using the formula:
A = b * y
where y is the depth of water in the channel
- Using the alternate depths, we have two possible cross-sectional areas:
A1 = b * 1.0 = b
A2 = b * 1.5 = 1.5b
- Since both the areas carry the same discharge, we can equate them:
A1 * V1 = A2 * V2
where V1 and V2 are the velocities of water in the two sections
- Since the channel is wide, we can assume that the velocity of water is same in both sections. Therefore, we can simplify the equation as:
A1 = A2
b = 1.5b
b = 0.67 m
- Therefore, the width of the channel is 0.67 m.
- Now, we can calculate the discharge intensity using either of the alternate depths and the width of the channel. Let's use the depth of 1.0 m.
- The cross-sectional area of the channel is:
A = b * y = 0.67 * 1.0 = 0.67 m^2
- The discharge intensity (q) can be calculated using the formula:
q = A * V
where V is the velocity of water in the channel.
- Since the velocity is not given, we cannot calculate the discharge intensity.

3. Conclusion

- The discharge intensity cannot be calculated without knowing the velocity of water in the channel.